1. Judea Pearl University of California Los Angeles (www.cs.ucla.edu/~judea) THE MATHEMATICS OF CAUSE AND EFFECT

2. Causality: Antiquity to robotics Modeling: Statistical vs. Causal Causal Models and Identifiability Inference to three types of claims: 1.Effects of potential interventions 2.Claims about attribution (responsibility) 3.Claims about direct and indirect effects OUTLINE

3. ANTIQUITY TO ROBOTICS “I would rather discover one causal relation than be King of Persia” Democritus (430-380 BC) Development of Western science is based on two great achievements: the invention of the formal logical system (in Euclidean geometry) by the Greek philosophers, and the discovery of the possibility to find out causal relationships by systematic experiment (during the Renaissance). A. Einstein, April 23, 1953

4. David Hume ( 1711–1776 )

5. HUME’S LEGACY HUME’S LEGACY 1.Analytical vs. empirical claims 2.Causal claims are empirical 3.All empirical claims originate from experience.

6. THE TWO RIDDLES OF CAUSATION l What empirical evidence legitimizes a cause-effect connection? l What inferences can be drawn from causal information? and how?

8. “Easy, man! that hurts!” The Art of Causal Mentoring

9. 1. How should a robot acquire causal information from the environment? 2. How should a robot process causal information received from its creator-programmer? OLD RIDDLES IN NEW DRESS

10. Input: 1. “If the grass is wet, then it rained” 2. “if we break this bottle, the grass will get wet” Output: “If we break this bottle, then it rained” CAUSATION AS A PROGRAMMER'S NIGHTMARE

11. CAUSATION AS A PROGRAMMER'S NIGHTMARE (Cont.) ( Lin, 1995) Input: 1. A suitcase will open iff both locks are open. 2. The right lock is open Query: What if we open the left lock? Output: The right lock might get closed.

12. THE BASIC PRINCIPLES Causation = encoding of behavior under interventions Interventions= surgeries on mechanisms Mechanisms = stable functional relationships = equations + graphs

13. TRADITIONAL STATISTICAL INFERENCE PARADIGM Data Inference Q(P) (Aspects of P ) P Joint Distribution e.g., Infer whether customers who bought product A would also buy product B. Q = P(B | A)

14. What happens when P changes? e.g., Infer whether customers who bought product A would still buy A if we were to double the price. FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES Probability and statistics deal with static relations Data Inference Q( P ) (Aspects of P ) P Joint Distribution P Joint Distribution change

15. FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES Note: P (v)  P (v | price = 2) P does not tell us how it ought to change e.g. Curing symptoms vs. curing diseases e.g. Analogy: mechanical deformation What remains invariant when P changes say, to satisfy P (price=2)=1 Data Inference Q( P ) (Aspects of P ) P Joint Distribution P Joint Distribution change

16. FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES (CONT) CAUSAL Spurious correlation Randomization Confounding / Effect Instrument Holding constant Explanatory variables STATISTICAL Regression Association / Independence “Controlling for” / Conditioning Odd and risk ratios Collapsibility 1.Causal and statistical concepts do not mix. 2. 3. 4.

17. CAUSAL Spurious correlation Randomization Confounding / Effect Instrument Holding constant Explanatory variables STATISTICAL Regression Association / Independence “Controlling for” / Conditioning Odd and risk ratios Collapsibility 1.Causal and statistical concepts do not mix. 4. 3.Causal assumptions cannot be expressed in the mathematical language of standard statistics. FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES (CONT) 2.No causes in – no causes out (Cartwright, 1989) statistical assumptions + data causal assumptions causal conclusions  }

18. 4.Non-standard mathematics: a)Structural equation models (Wright, 1920; Simon, 1960) b)Counterfactuals (Neyman-Rubin (Y x ), Lewis (x Y)) CAUSAL Spurious correlation Randomization Confounding / Effect Instrument Holding constant Explanatory variables STATISTICAL Regression Association / Independence “Controlling for” / Conditioning Odd and risk ratios Collapsibility 1.Causal and statistical concepts do not mix. 3.Causal assumptions cannot be expressed in the mathematical language of standard statistics. FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES (CONT) 2.No causes in – no causes out (Cartwright, 1989) statistical assumptions + data causal assumptions causal conclusions  }

19. 1.Every exercise of causal analysis must rest on untested, judgmental causal assumptions. 2.Every exercise of causal analysis must invoke non-standard mathematical notation. FROM STATISTICAL TO CAUSAL ANALYSIS: 2. THE MENTAL BARRIERS

20. TWO PARADIGMS FOR CAUSAL INFERENCE Observed: P(X, Y, Z,...) Conclusions needed: P(Y x =y), P(X y =x | Z=z)... How do we connect observables, X,Y,Z,… to counterfactuals Y x, X z, Z y,… ? N-R model Counterfactuals are primitives, new variables Super-distribution P*(X, Y,…, Y x, X z,…) X, Y, Z constrain Y x, Z y,… Structural model Counterfactuals are derived quantities Subscripts modify a data-generating model

21. “SUPER” DISTRIBUTION IN N-R MODEL X0 0 0 1 X0 0 0 1 Y 0 1 0 0 Y 0 1 0 0 Y x=0 0 1 1 1 Z0 1 0 0 Z0 1 0 0 Y x=1 1 0 0 0 X z=0 0 1 0 0 X z=1 0 0 1 1 X y=0  0  1  1  0  Uu1 u2 u3 u4 Uu1 u2 u3 u4 inconsistency: x = 0  Y x=0 = Y Y = xY 1 + (1-x) Y 0

22. Data Inference Q(M) (Aspects of M ) Data Generating Model M – Oracle for computing answers to Q ’s. e.g., Infer whether customers who bought product A would still buy A if we were to double the price. Joint Distribution THE STRUCTURAL MODEL PARADIGM

23. Z Y X INPUTOUTPUT FAMILIAR CAUSAL MODEL ORACLE FOR MANIPILATION

24. STRUCTURAL CAUSAL MODELS Definition: A structural causal model is a 4-tuple  V,U, F, P(u) , where V = {V 1,...,V n } are observable variables U = {U 1,...,U m } are background variables F = {f 1,..., f n } are functions determining V, v i = f i (v, u) P(u) is a distribution over U P(u) and F induce a distribution P(v) over observable variables

25. U (Court order) D (Death) B (Riflemen) C (Captain) A CAUSAL MODELS AT WORK (The impatient firing-squad)

26. CAUSAL MODELS AT WORK CAUSAL MODELS AT WORK(Glossary) U:Court orders the execution C:Captain gives a signal A:Rifleman-A shoots B:Rifleman-B shoots D:Prisoner dies =:Functional Equality (new symbol) U D B C A C=U A=CB=C D=A  B

27. SENTENCES TO BE EVALUATED S1. prediction:  A   D S2. abduction:  D   C S3. transduction: A  B S4. action:  C  D A S5. counterfactual: D  D {  A} S6. explanation: Caused(A, D) U D B C A

28. STANDARD MODEL FOR STANDARD QUERIES S1. (prediction): If rifleman-A shot, the prisoner is dead, A  D S2. (abduction): If the prisoner is alive, then the Captain did not signal,  D   C S3. (transduction): If rifleman-A shot, then B shot as well, A  B U D B C A iff  OR

29. WHY CAUSAL MODELS? GUIDE FOR SURGERY S4. (action): If the captain gave no signal and Mr. A decides to shoot, the prisoner will die:  C  D A, and B will not shoot:  C   B A U D B C A

30. WHY CAUSAL MODELS? GUIDE FOR SURGERY S4. (action): If the captain gave no signal and Mr. A decides to shoot, the prisoner will die:  C  D A, and B will not shoot:  C   B A U D B C A  TRUE

31. U D B C A AbductionActionPrediction S5.If the prisoner is dead, he would still be dead if A were not to have shot. D  D  A 3-STEPS TO COMPUTING COUNTERFACTUALS U D B C A FALSE TRUE U D B C A FALSE TRUE

32. U D B C A Abduction P(S5).The prisoner is dead. How likely is it that he would be dead if A were not to have shot. P(D  A |D) = ? COMPUTING PROBABILITIES OF COUNTERFACTUALS Action TRUE Prediction U D B C A FALSE P(u|D) P(D  A |D ) P(u)P(u) U D B C A FALSE P(u|D)

33. CAUSAL MODEL (FORMAL) M = U - Background variables V - Endogenous variables F - Set of functions {U  V \V i  V i } v i =f i (pa i, u i ) Submodel: M x =, representing do(x) F x = Replaces equation for X with X=x Actions and Counterfactuals: Y x (u) = Solution of Y in M x or P(y | do(x)) P(Y x =y)

34. WHY COUNTERFACTUALS? Action queries are triggered by (modifiable) observations, demanding abductive step, i.e., counterfactual processing. E.g., Troubleshooting Observation:The output is low Action query:Will the output get higher – if we replace the transistor? Counterfactual query:Would the output be higher – had the transistor been replaced?

35. APPLICATIONS 1. Predicting effects of actions and policies 2. Learning causal relationships from assumptions and data 3. Troubleshooting physical systems and plans 4. Finding explanations for reported events 5. Generating verbal explanations 6. Understanding causal talk 7. Formulating theories of causal thinking

36. CAUSAL MODELS AND COUNTERFACTUALS Definition: The sentence: “ Y would be y (in situation u ), had X been x,” denoted Y x (u) = y, means: The solution for Y in a mutilated model M x, (i.e., the equations for X replaced by X = x ) with input U=u, is equal to y. Joint probabilities of counterfactuals: The super-distribution P * is derived from M. Parsimonious, consistent, and transparent

37. AXIOMS OF CAUSAL COUNTERFACTUALS 1.Definiteness 2.Uniqueness 3.Effectiveness 4.Composition 5.Reversibility Y would be y, had X been x (in state U = u )

38. GRAPHICAL – COUNTERFACTUALS SYMBIOSIS Every causal graph expresses counterfactuals assumptions, e.g., Assumptions are guaranteed consistency. Assumptions are readable from the graph. 1.Missing arrows X  Z 2. Missing arcs YZ X  Y  Z

39. RULES OF CAUSAL CALCULUS Rule 1: Ignoring observations P(y | do{x}, z, w) = P(y | do{x}, w) Rule 2: Action/observation exchange P(y | do{x}, do{z}, w) = P(y | do{x},z,w) Rule 3: Ignoring actions P(y | do{x}, do{z}, w) = P(y | do{x}, w)

40. DERIVATION IN CAUSAL CALCULUS Smoking Tar Cancer P (c | do{s}) =  t P (c | do{s}, t) P (t | do{s}) =  s   t P (c | do{t}, s) P (s | do{t}) P(t |s) =  t P (c | do{s}, do{t}) P (t | do{s}) =  t P (c | do{s}, do{t}) P (t | s) =  t P (c | do{t}) P (t | s) =  s  t P (c | t, s) P (s) P(t |s) =  s   t P (c | t, s) P (s | do{t}) P(t |s) Probability Axioms Rule 2 Rule 3 Rule 2 Genotype (Unobserved)

41. THE BACK-DOOR CRITERION Graphical test of identification P(y | do(x)) is identifiable in G if there is a set Z of variables such that Z d -separates X from Y in G x. Z6Z6 Z3Z3 Z2Z2 Z5Z5 Z1Z1 X Y Z4Z4 Z6Z6 Z3Z3 Z2Z2 Z5Z5 Z1Z1 X Y Z4Z4 Z Moreover, P(y | do(x)) =   P(y | x,z) P(z) (“adjusting” for Z ) z GxGx G

42. do -calculus is complete Complete graphical criterion for identifying causal effects (Shpitser and Pearl, 2006). Complete graphical criterion for empirical testability of counterfactuals (Shpitser and Pearl, 2007). RECENT RESULTS ON IDENTIFICATION

43. DETERMINING THE CAUSES OF EFFECTS (The Attribution Problem) Your Honor! My client (Mr. A) died BECAUSE he used that drug.

44. DETERMINING THE CAUSES OF EFFECTS (The Attribution Problem) Your Honor! My client (Mr. A) died BECAUSE he used that drug. Court to decide if it is MORE PROBABLE THAN NOT that A would be alive BUT FOR the drug! P (? | A is dead, took the drug) > 0.50 PN =

45. THE PROBLEM Semantical Problem: 1.What is the meaning of PN(x,y): “Probability that event y would not have occurred if it were not for event x, given that x and y did in fact occur.”

46. THE PROBLEM Semantical Problem: 1.What is the meaning of PN(x,y): “Probability that event y would not have occurred if it were not for event x, given that x and y did in fact occur.” Answer: Computable from M

47. THE PROBLEM Semantical Problem: 1.What is the meaning of PN(x,y): “Probability that event y would not have occurred if it were not for event x, given that x and y did in fact occur.” 2.Under what condition can PN(x,y) be learned from statistical data, i.e., observational, experimental and combined. Analytical Problem:

48. TYPICAL THEOREMS (Tian and Pearl, 2000) Bounds given combined nonexperimental and experimental data Identifiability under monotonicity (Combined data) corrected Excess-Risk-Ratio

49. CAN FREQUENCY DATA DECIDE LEGAL RESPONSIBILITY? Nonexperimental data: drug usage predicts longer life Experimental data: drug has negligible effect on survival ExperimentalNonexperimental do(x) do(x) x x Deaths (y) 16 14 2 28 Survivals (y) 984 986 998 972 1,0001,0001,0001,000 1.He actually died 2.He used the drug by choice Court to decide (given both data): Is it more probable than not that A would be alive but for the drug? Plaintiff: Mr. A is special.

50. SOLUTION TO THE ATTRIBUTION PROBLEM WITH PROBABILITY ONE 1  P(y x | x,y)  1 Combined data tell more that each study alone

51. EFFECT DECOMPOSITION What is the semantics of direct and indirect effects? What are their policy-making implications? Can we estimate them from data? Experimental data?

52. 1.Direct (or indirect) effect may be more transportable. 2.Indirect effects may be prevented or controlled. 3.Direct (or indirect) effect may be forbidden WHY DECOMPOSE EFFECTS? Pill Thrombosis Pregnancy +  + Gender Hiring Qualification

53. TOTAL, DIRECT, AND INDIRECT EFFECTS HAVE SIMPLE SEMANTICS IN LINEAR MODELS XZ Y c a b z = bx +  1 y = ax + cz +  2 a + bc bc a

54. z = f (x,  1 ) y = g (x, z,  2 ) XZ Y SEMANTICS BECOMES NONTRIVIAL IN NONLINEAR MODELS (even when the model is completely specified) Dependent on z ? Void of operational meaning?

55. z = f (x,  1 ) y = g (x, z,  2 ) XZ Y THE OPERATIONAL MEANING OF DIRECT EFFECTS “Natural” Direct Effect of X on Y: The expected change in Y per unit change of X, when we keep Z constant at whatever value it attains before the change. In linear models, NDE = Controlled Direct Effect

56. ``The central question in any employment-discrimination case is whether the employer would have taken the same action had the employee been of different race (age, sex, religion, national origin etc.) and everything else had been the same’’ [ Carson versus Bethlehem Steel Corp. (70 FEP Cases 921, 7 th Cir. (1996))] x = male, x = female y = hire, y = not hire z = applicant’s qualifications LEGAL DEFINITIONS TAKE THE NATURAL CONCEPTION (FORMALIZING DISCRIMINATION) NO DIRECT EFFECT

57. z = f (x,  1 ) y = g (x, z,  2 ) XZ Y THE OPERATIONAL MEANING OF INDIRECT EFFECTS “Natural” Indirect Effect of X on Y: The expected change in Y when we keep X constant, say at x 0, and let Z change to whatever value it would have under a unit change in X. In linear models, NIE = TE - DE

58. POLICY IMPLICATIONS OF INDIRECT EFFECTS f GENDERQUALIFICATION HIRING What is the direct effect of X on Y? The effect of Gender on Hiring if sex discrimination is eliminated. indirect XZ Y IGNORE

59. SEMANTICS AND IDENTIFICATION OF NESTED COUNTERFACTUALS Consider the quantity Given  M, P(u) , Q is well defined Given u, Z x * (u) is the solution for Z in M x *, call it z is the solution for Y in M xz Can Q be estimated from data?

60. Example: Theorem: If there exists a set W such that GRAPHICAL CONDITION FOR EXPERIMENTAL IDENTIFICATION OF AVERAGE NATURAL DIRECT EFFECTS

61. Corollary 3: The average natural direct effect in Markovian models is identifiable from nonexperimental data, and it is given by IDENTIFICATION IN MARKOVIAN MODELS X Z Y

62. Theorem 5: The total, direct and indirect effects obey The following equality In words, the total effect (on Y) associated with the transition from x * to x is equal to the difference between the direct effect associated with this transition and the indirect effect associated with the reverse transition, from x to x *. RELATIONS BETWEEN TOTAL, DIRECT, AND INDIRECT EFFECTS

63. Y Z X W x*x* z * = Z x* (u) Nonidentifiable even in Markovian models GENERAL PATH-SPECIFIC EFFECTS (Def.) Y Z X W Form a new model,, specific to active subgraph g Definition: g -specific effect

64. EFFECT DECOMPOSITION SUMMARY Graphical conditions for estimability from experimental / nonexperimental data. Useful in answering new type of policy questions involving mechanism blocking instead of variable fixing. Graphical conditions hold in Markovian models

65. CONCLUSIONS Structural-model semantics, enriched with logic and graphs, provides: Complete formal basis for causal reasoning Powerful and friendly causal calculus Lays the foundations for asking more difficult questions: What is an action? What is free will? Should robots be programmed to have this illusion?